Nonlinear transport phenomena: models, method of solving and unusual features (3)

1. Nonlinear transport phenomena: models, method of solving and unusual features: Lecture 3 Vsevolod Vladimirov AGH University of Science and technology, Faculty of Applied Mathematics ´ Krakow, August 6, 2010 KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 1 / 34

2. Traveling wave solutions supported by GBE We are interested in the following generalization of the BE, called convection-diﬀusion - reaction equation: τ ut t + ut + u ux = ν [un ux ]x + f (u) (1) Our aim is to study the wave patterns, i.e. physically meaningful traveling wave (TW) solutions, having the form u(t, x) = U (ξ), ξ = x − V t. We are specially interest in the existence of solitons, compactons, some other solutions related to them KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 2 / 34

8. Solitons and compactons are solitary waves, moving with constant velocity V without change of their shape. The main diﬀerence between them is seen on the graphs shown below: A Figure: Graph of the KdV soliton U (ξ) = Cosh2 [B ξ] KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 3 / 34

9. Figure: Graph of the Rosenau-Hyman compacton ACos2 [B ξ], if |ξ| < 2 π, U (ξ) = 0 otherwise KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 4 / 34

10. So how can we distinguish the solitary wave solutions and compactons within the set of TW solutions? To answer this question, we restore to the geometric interpretation of these solutions. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 5 / 34

11. So how can we distinguish the solitary wave solutions and compactons within the set of TW solutions? To answer this question, we restore to the geometric interpretation of these solutions. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 5 / 34

12. Soliton on the phase plane of factorized system Both solitons and compactons are the TW solutions, having the form u(t, x) = U (ξ), with ξ = x − V t. We give the geometric interpretation of the soliton by considering the Korteveg-de Vries (KdV) equation ut + u ux + uxxx = 0 Inserting the TW ansatz into the KdV equation, we get: d ¨ U 2 (ξ) U (ξ) + − V U (ξ) = 0. dξ 2 This equation is equivalent to the following dynamical system: KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 6 / 34

16. ˙ U (ξ) = −W (ξ), 2 (ξ) (2) W (ξ) = U 2 − V U (ξ) ˙ Lemma 1.The system (2) is a Hamiltonian system, with W2 U3 U2 H(U, W ) = + −V = Ekin (W ) + Epot (U ). (3) 2 6 2 Lemma 2.Hamiltonian function H(U, W ) remains constant along a trajectory of the dynamical system (2). Theorem 1. 1 All stationary points of the system (2) are placed at the 3 2 extremal points of the function Epot (U ) = U − V U ; 6 2 local maximum of Epot (U ) correspond to a saddle point, while the local minimum corresponds to a center; all possible coordinates of the phase trajectory corresponding to H(U, W ) = C satisfy the inequalities C − Epot (U ) ≥ 0. 1 Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966 KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 7 / 34

23. U3 U2 Graph of the function Epot (U ) = 6 −V 2 Phase portrait of the system (2) KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 8 / 34

26. Statement 1. The only trajectory that could correspond to the soliton solution is the closed loop bi-asymptotic to the saddle point. Statement 2. The solitary wave solution U (ξ) is nonzero for any ξ ∈ R. ˙ ˙ Proof. The linearized system U = −W, W = −V U is ¨ equivalent to the single equation U = V U . Solution to this equation, that can approach zero, must have the form U (ξ) = C e± V ξ . It is obvious, that the function U (ξ) is nonzero for any nonzero ξ, and can attain zero when ξ → ± ∞. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 9 / 34

31. Compactons on the phase plane of factorized system Let us discuss the compacton TW solutions, basing on Rosenau-Hyman equation ut + u2 x + u2 xxx = 0. (4) Inserting the TW ansatz u(t, x) = U (ξ), ξ = x − V t into (4), we get, after some manipulation, the dynamical system dU dT = −2 U 2 W, dW , (5) dT = U −V U + U 2 + 2 W 2 d d where dT = 2 U2 dξ. Lemma 3.The system (5) is a Hamiltonian system, with 1 4 V 3 H= U − U + U 2 W 2. 4 3 KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 10 / 34

34. Graph of the function Epot (U ) Phase portrait of the system (5) KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 11 / 34

37. Statement 3. The solution of to (5) corresponding to the homoclinic loop has a compact support. Sketch of the proof. It can be shown, that the saddle separatrices enter (leave) the origin forming right angle with the horizontal axis.Therefore, in proximity of the origin W ≈ −C1 U α with 0 < α < 1. From this we get the asymptotic equation −W = d U = C1 U α . It is evident, thus, that the dξ asymptotic solution takes the form U (ξ) ≈ C2 (ξ − ξ0 )γ , 1 < γ = 1−α . So the function U (ξ) 1 approaches zero as ξ → ξ0 ± 0. So we deal in fact with the glued (generalized) solution, and its nonzero part corresponds to the homoclinic loop. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34

42. Very important conclusion The localized wave patterns, such as solitary waves and compactons are represented in the phase plane of the factorized system by the HOMOCLINIC LOOP, i.e. the phase trajectory bi-asymptotic to a saddle point. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 13 / 34

43. Factorization of the modeling system Let’s return to the generalized C-R-D equation: α utt + ut + u ux − κ (un ux )x = f (u). (6) Inserting the TW ansatz u(t, x) = U (ξ) ≡ U (x − V t) into this system, we get: ˙ ∆(U )U = ∆(U ) W, (7) ˙ ∆(U )W = − f (U ) + κn U n−1 2 W + (V − U ) W , where ∆(U ) = κ U n − α V 2 . KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 14 / 34

48. The time-delayed C − R − D equation is of dissipative type. Therefore: the factorized system is not Hamiltonian; the homoclinic trajectory will appear at the speciﬁc values of the parameters!!! KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 15 / 34

51. Our further strategy: we choose speciﬁc function f (u) such that the factorized system will have at least two stationary points (U0 , 0) and (U1 , 0) with 0 ≤ U0 < U1 ; we state the condition for which the stationary point (U1 , 0) becomes a center; we state, the conditions which guarantee that the other point (U0 , 0) simultaneously is a saddle; we analyze the appearance of the limit cycle with zero radius at the point (U1 , 0); we check numerically the possibility of homoclinic bifurcation appearance. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 16 / 34

57. Further assumptions: We assume that f (U ) = (U − U0 )m (U − U1 ) ψ(U ), U1 > U0 ≥ 0, where ψ(U )|<U0 , U1 > = 0. Under these assumption our system has two stationary points (U0 , 0) and (U1 , 0) lying on the horizontal axis of the phase space (U, W ), and no any other stationary point inside the segment < U0 , U1 >. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 17 / 34

58. Creation of a stable limit cycle Theorem 1. 1. If the following conditions hold ∆(U1 ) · ψ(U1 ) ≡ κ U1 − α V 2 ψ(U1 ) > 0, n (8) and n−1 |∆(U1 )| ϕ(U1 ) + κ n U1 |ϕ(U1 )| > 0, ˙ (9) where ϕ(U ) = (U − U0 )m ψ(U ), ∆(U ) = κ U n − α V 2 then in vicinity of the stationary point (U1 , 0) a stable limit cycle with zero radius is created, when the wave pack velocity V approaches the bifurcation value Vcr1 = U1 . 2. Under these conditions, the other stationary point (U0 , 0) is a topological saddle, or, at least, contains a saddle sector in the half-plane U > U0 . KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 18 / 34

61. A peculiarity of our factorized system ˙ ∆(U )U = ∆(U ) W, (10) ˙ ∆(U )W = − f (U ) + κn U n−1 W 2 + (V − U ) W , where f (U ) = (U − U1 ) (U − U0 )m ψ(U ), is the presence of the line of singularities ∆(U ) = κ U n − α V 2 = 0, moving along the phase plane, as the bifurcation parameter V is changed. This creates an extra mechanism of the limit cycle destruction, diﬀerent from the homoclinic bifurcation. But it is just the singular line, which makes possible the presence of such TW solutions, as compactons, shock fronts and the peakons. A necessary condition of their appearance reads as follows: the singular line must contain the topological saddle at the moment of the homoclinic bifurcation! KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 19 / 34

65. To what kind of localized solution corresponds the homoclinic loop? Asymptotic study of the dynamical system, corresponding to source equation α utt + ut + u ux − κ (un ux )x = (u − U0 )m (u − U1 ) ψ(u) (11) enable to state that: homoclinic loop corresponds to the compactly-supported TW if 0 < m < 1, and n ∈ N, and U∗ = U0 , where U∗ ⇔ ∆(U ) = 0 ; homoclinic loop corresponds to the soliton-like TW if m = 1, n ∈ N and U∗ < U0 ; homoclinic loop corresponds to a semi-compacton (or shock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0 . KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 20 / 34

69. Figure: Vicinity of the origin for various combinations of the parameters m, n KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 21 / 34

70. Numerical investigation of factorized system Numerical simulations of the system (10) were carried out with κ = 1, U1 = 3, U0 = 1. The remaining parameters varied from one case to another. We discuss the results concerning the details of the phase portraits in terms of the reference frame (X, W ) = (U − U0 , W ). The localized wave patterns are presented in ”physical” coordinates (ξ, U ), where ξ = x − V t is the TW coordinate KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 22 / 34

73. Figure: Homoclinic solution of the system (10) with ϕ(U ) = (U − U0 )1/2 (m = 1/2)(left) and the corresponding compactly supported TW solution to Eq. (11) (right), obtained for n = 1, α = 0.12, Vcr2 ∼ 2.68687 and U∗ − U0 = −0.133684 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 23 / 34

74. Figure: Homoclinic solution of the system (10) with ϕ(U ) = (U − U0 )1/2 (m = 1/2) (left) and the corresponding TW solution to Eq. (11) (right), obtained for n = 1, α = 0.13827, Vcr2 ∼ 2.68892 and U∗ − U0 = 0.99973 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 24 / 34

75. Figure: Homoclinic solution of the system (10) with ϕ(U ) = (U − U0 )1 (m = 1) (left), the corresponding tandem of well-localized soliton-like solutions to Eq. (11) (center), and the soliton-like solution (right), obtained for n = 1, α = 0.06, Vcr2 ∼ 2.65795 and U∗ − U0 = −0.576119 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 25 / 34

76. Figure: Homoclinic solution of the system (10) with ϕ(U ) = (U − U0 )1 (m = 1) (left), the corresponding tandem od solitary wave solutions to Eq. (11) (center) and a single solitary wave solution (right), obtained for n = 1, α = 0.142, Vcr2 ∼ 2.65489 and U∗ − U0 = 0.000878617 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 26 / 34

77. Figure: Tandems od shock-like solutions to Eq. (11), corresponding to ϕ(U ) = (U − U0 )1 (m = 1), U∗ ≈ U0 , n = 2 (left), n = 3 (center), and n = 4 (right) KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 27 / 34

78. Figure: Shock-like solutions to Eq. (11), corresponding to ϕ(U ) = (U − U0 )3 (m = 3), U∗ ≈ U0 , n = 1 (left), n = 3 (center), and n = 4 (right) KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 28 / 34

79. Figure: Shock-like solutions to Eq. (11), corresponding to ϕ(U ) = (U − U0 )m , U∗ ≈ U0 , n = 4, m = 1 (left), m = 2 (center), and m = 3 (right) KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 29 / 34

80. Figure: Periodic solution of the system (10) with ϕ(U ) = −(U − U0 )1 (m = 1) (left) and the corresponding tandem of generalized peak-like solutions to Eq. (11) (right), obtained for n = 1, α = 0.552, Vcr2 ∼ 3.00593 and U∗ − U1 = 1.98765 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 30 / 34

81. Figure: Homoclinic solution of the system (10) with ϕ(U ) = −(U − U0 )1/2 (m = 1/2) (left) and the corresponding tandem of generalized peak-like solutions to Eq. (11) (right), obtained for n = 1, α = 0.562, Vcr2 ∼ 3.14497 and U∗ − U1 = 2.55863 = KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 31 / 34

82. Summary 1. The time-delayed C − R − D system possesses a large variety of the localized wave patterns. 2. The type of the pattern is strongly depend on the values of the parameters 3. An open question is the analytical description of the localized wave patterns within the given model. 4. Another open question is the stability and the attractive features of the localized wave patterns. KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 32 / 34

86. THANKS FOR YOUR ATTENTION KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 33 / 34

87. Delayed C-R-D equation can be formally obtained if we consider the integro-diﬀerential equation t 1 t−t ut = exp[− ] [κ (un ux )x − u ux + f (u)] (t , x) α −∞ α instead of ut = κ (un ux )x − u ux + f (u). Integrating the integro-diﬀerential equation w.r.t. temporal variable, w obtain the target equation α ut t + ut = κ (un ux )x − u ux + f (u). KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 34 / 34

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